The Galois Complexity of Graph Drawing Michael J. Bannister William E. Devanny David Eppstein Michael Goodrich
Overview Motivation Galois theory Models of computation Results Undrawable graphs!
Overview Motivation Galois theory Models of computation Results Undrawable graphs! For some definition of undrawable
Motivation x max c T x c Simplex vs Interior Point
Motivation max c T x x c Simplex Methods
Motivation max c T x x c Simplex Methods
Motivation x max c T x c Simplex Methods
Motivation x max c T x c Interior Point Methods
Motivation max c T x x c Interior Point Methods
Motivation x max c T x c Interior Point Methods
Motivation x max c T x c Interior Point Methods
Motivation x max c T x c Interior Point Methods
Motivation Symbolic vs. Numerical algorithms Symbolic - manipulate mathematical expressions to obtain an exact answer for a problem Simplex method Numerical - iteratively walk towards the answer, improving an approximate answer with each step Interior point method
Force Directed Graph Drawing (Fruchterman and Reingold) Neighbors: f a ( d ) = d 2 /k All pairs: f r ( d ) = k 2 /d When the total force at each vertex is zero, we are at F. and R. equilbrium Also disqualify unstable/degenerate solutions
Force Directed Graph Drawing (Fruchterman and Reingold) Neighbors: f a ( d ) = d 2 /k All pairs: f r ( d ) = k 2 /d When the total force at each vertex is zero, we are at F. and R. equilbrium Also disqualify unstable/degenerate solutions
Force Directed Graph Drawing (Fruchterman and Reingold) Neighbors: f a ( d ) = d 2 /k All pairs: f r ( d ) = k 2 /d When the total force at each vertex is zero, we are at F. and R. equilbrium Also disqualify unstable/degenerate solutions
Force Directed Graph Drawing (Fruchterman and Reingold) Neighbors: f a ( d ) = d 2 /k All pairs: f r ( d ) = k 2 /d When the total force at each vertex is zero, we are at F. and R. equilbrium Also disqualify unstable/degenerate solutions
Force Directed Graph Drawing (Fruchterman and Reingold) Neighbors: f a ( d ) = d 2 /k length k All pairs: f r ( d ) = k 2 /d When the total force at each vertex is zero, we are at F. and R. equilbrium Also disqualify unstable/degenerate solutions
Force Directed Graph Drawing (Fruchterman and Reingold) Neighbors: f a ( d ) = d 2 /k All pairs: f r ( d ) = k 2 /d When the total force at each vertex is zero, we are at F. and R. equilbrium Also disqualify unstable/degenerate solutions
Force Directed Graph Drawing (Fruchterman and Reingold) Neighbors: f a ( d ) = d 2 /k All pairs: f r ( d ) = k 2 /d When the total force at each vertex is zero, we are at F. and R. equilbrium Also disqualify unstable/degenerate solutions
Force Directed Graph Drawing (Fruchterman and Reingold) Neighbors: f a ( d ) = d 2 /k length k All pairs: f r ( d ) = k 2 /d When the total force at each vertex is zero, we are at F. and R. equilbrium Also disqualify unstable/degenerate solutions
Motivation - Graph Drawing Many problems only have numerical algorithms Fruchterman-Reingold Kamada-Kawai Spectral methods Circle packings Why?
Motivation - Graph Drawing Many problems only have numerical algorithms Fruchterman-Reingold Kamada-Kawai Spectral methods Circle packings Why? Galois theory!
Solving polynomials Quadratics √ ax 2 + bx + c = 0 b 2 − 4 ac x = − b ± ⇒ 2 a Cubics ax 3 + bx 2 + cx + d = 0 b Substitute x = t − ⇒ 3 a t 3 + pt + q = 0 p Substitute t = w − 3 w w 6 + qw 3 − p 3 27 = 0 Quadratic in w 3
Solving polynomials Quartic ax 4 + bx 3 + cx 2 + dx + e = 0 ⇒ Still has a symbolic solution Very messy Quintic ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0 ⇒ ?
Galois - a short biography Born in France in 1811 Mathematician First to use group as a technical term Worked on polynomial equations Political activist Was expelled for his political opinions Imprisoned for threatening the King’s life Is shot and killed in the duel in 1832 Showed there is no quintic formula the night before
Galois Theory Draws a connection between groups and roots of polynomials where the group encodes the expressibility of the roots If the Galois group for a polynomial contains S 5 as a subgroup, then the roots cannot be written using radicals Written using radicals? √ φ = 1 . 618 ... = 1+ 5 π = 3 . 14159 . . . 2
Models of Computation Algebraic computation tree A model in which each node makes a decision or computes a value using standard arithmetic functions of previous values x = 5 − y x > y ? True False Accept Reject
Models of Computation Quadratic computation tree Radical computation tree Bounded degree root computation tree
Models of Computation Quadratic computation tree An algebraic computation tree with square roots and complex conjugation Radical computation tree Bounded degree root computation tree
Models of Computation Quadratic computation tree An algebraic computation tree with square roots and complex conjugation Radical computation tree An algebraic computation tree with k th roots and complex conjugation for any integer k Bounded degree root computation tree
Models of Computation Quadratic computation tree An algebraic computation tree with square roots and complex conjugation Radical computation tree An algebraic computation tree with k th roots and complex conjugation for any integer k Bounded degree root computation tree An algebraic computation tree with taking roots of bounded degree polynomials and complex conjugation
Models of Computation Quadratic computation tree An algebraic computation tree with square roots and complex conjugation Compass and straightedge model Radical computation tree An algebraic computation tree with k th roots and complex conjugation for any integer k Bounded degree root computation tree An algebraic computation tree with taking roots of bounded degree polynomials and complex conjugation
Approach F & R Layout Graph Drawing Layout Polynomial Galois Group Expressibility in a Symbolic Model
Approach F & R Layout Graph Drawing Layout Polynomial p ( x ) = x n + . . . Galois Group Expressibility in a Symbolic Model
Approach F & R Layout Graph Drawing Layout Polynomial p ( x ) = x n + . . . Galois Group S 5 Expressibility in a Symbolic Model
Approach F & R Layout Graph Drawing Layout Polynomial p ( x ) = x n + . . . Galois Group S 5 Expressibility in a Symbolic Model Cannot draw in a Radical Computation Tree
Approach F & R Layout Lots of variables Graph Drawing Layout System of polynomials Polynomial p ( x ) = x n + . . . Galois Group S 5 Expressibility in a Symbolic Model Cannot draw in a Radical Computation Tree
Approach F & R Layout Lots of variables Graph Drawing Layout System of polynomials ⇒ p ( x ) may have high degree Polynomial p ( x ) = x n + . . . Galois Group S 5 Expressibility in a Symbolic Model Cannot draw in a Radical Computation Tree
Approach F & R Layout Lots of variables Graph Drawing Layout System of polynomials ⇒ p ( x ) may have high degree Polynomial p ( x ) = x n + . . . Galois Group S 5 Expressibility in a Symbolic Model Cannot draw in a Radical Computation Tree
Approach F & R Layout Lots of variables Graph Drawing Layout System of polynomials ⇒ p ( x ) may have high degree Polynomial Exploit symmetry to reduce degree p ( x ) = x n + . . . Galois Group S 5 Expressibility in a Symbolic Model Cannot draw in a Radical Computation Tree
Undrawable Graphs Bounded degree root computation trees Fruchterman-Reingold Circle packings Kamada-Kawai Spectral graph drawings
Undrawable Graphs Bounded degree root computation trees p -cycles Fruchterman-Reingold Circle packings Kamada-Kawai Spectral graph drawings
Undrawable Graphs Bounded degree root computation trees p -cycles p -bipyramid Fruchterman-Reingold Circle packings Kamada-Kawai Spectral graph drawings
Undrawable Graphs Radical computation trees Fruchterman-Reingold Kamada-Kawai Multidimensional scaling
Undrawable Graphs Radical computation trees Fruchterman-Reingold Kamada-Kawai Multidimensional scaling
Undrawable Graphs Radical computation trees Fruchterman-Reingold Kamada-Kawai Multidimensional scaling
Undrawable Graphs Radical computation trees Fruchterman-Reingold Kamada-Kawai Multidimensional scaling
Undrawable Graphs Radical computation trees Spectral graph drawings (Laplacian matrix) (Adjacency/Transition matrix)
Undrawable Graphs Radical computation trees Circle packings
Summary Lots of graph drawing uses numerical algorithms Why no symbolic algorithms? Galois theory! Graph drawing coordinates cannot be computed using radicals
Summary Lots of graph drawing uses numerical algorithms Why no symbolic algorithms? Galois theory! Graph drawing coordinates cannot be computed using radicals Open Questions Other graph drawing problems with no symbolic algorithms? Problems with arbitrarily high S n ?
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